### Table of Contents

## Closure algebras

Abbreviation: **CloA**

### Definition

A \emph{closure algebra} is a modal algebra $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,\diamond\rangle$ such that

$\diamond$ is \emph{closure operator}: $x\le \diamond x$, $\diamond\diamond x=\diamond x$

Remark: Closure algebras provide algebraic models for the modal logic S4. The operator $\diamond$ is the \emph{possibility operator}, and the \emph{necessity operator} $\Box$ is defined as $\Box x=\neg\diamond\neg x$.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be closure algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\diamond$:

$h(\diamond x)=\diamond h(x)$

### Examples

Example 1: $\langle P(X),\cup,\emptyset,\cap,X,-,cl\rangle$, where $X$ is any topological space and $cl$ is the closure operator associated with $X$.

### Basic results

### Properties

Classtype | variety |
---|---|

Equational theory | decidable |

Quasiequational theory | decidable |

First-order theory | undecidable |

Locally finite | no |

Residual size | unbounded |

Congruence distributive | yes |

Congruence modular | yes |

Congruence n-permutable | yes, $n=2$ |

Congruence regular | yes |

Congruence uniform | yes |

Congruence extension property | yes |

Definable principal congruences | yes |

Equationally def. pr. cong. | yes |

Discriminator variety | no |

Amalgamation property | yes |

Strong amalgamation property | yes |

Epimorphisms are surjective | yes |

### Finite members

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f(2)= &

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\end{array}$